Download - 5/2/2007Cohen Group Meeting1 Luttinger Liquids Kevin Chan Cohen Group Meeting May 2, 2007
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5/2/2007 Cohen Group Meeting 1
Luttinger Liquids
Kevin Chan
Cohen Group Meeting
May 2, 2007
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5/2/2007 Cohen Group Meeting 2
Introduction
• Luttinger Liquid: system of 1D interacting fermions
• Reasons for interest:– Strong correlations– Real (quasi-)1D systems available
experimentally– Exactly solvable
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5/2/2007 Cohen Group Meeting 3
Fermi vs. Luttinger Liquid
• Fermi Liquid– Landau– Elementary excitations
are quasiparticles (fermions)
– 1-1 correspondence between quasiparticles and excitations of FEG
– Weak correlations
• Luttinger Liquid– Tomonaga, Luttinger– Elementary excitations
are collective excitations (bosons)
– Strong correlations
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Luttinger Model
• Noninteracting Hamiltonian:
• Second quantization:• Spinless fermions• Two species• Linear dispersion
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Bosonization• Density operators:
• Commutation relations:
• Boson operators:
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Interaction Hamiltonian
• Scattering of same species:
• Different species:
Note: scattering must conserve momentum
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5/2/2007 Cohen Group Meeting 7
Solution of Model with Spin
• Now include spin:
• Define operators that obey Boson commutation relations:
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Spin-Charge Separation• Hamiltonian separates into charge and spin parts:
• Diagonalize charge density part:
• For delta function potential:Spin and charge have different speeds – spin-charge separation!
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5/2/2007 Cohen Group Meeting 9
1D versus 3D• Luttinger Liquids:
– Charge and spin density waves (bosons)– Can show that Fermi surface is unstable to perturbation
• Consider particle-hole excitation spectrum:
1D 3D
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5/2/2007 Cohen Group Meeting 10
Experiment: Carbon Nanotubes (ropes)• For LL, conductance should follow power law:
Bockrath et al., Nature
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5/2/2007 Cohen Group Meeting 11
CNT Junctions
• Conductance measurements for two joined CNT segments
I II Yao et al., Nature
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Spin-Charge Separation in Nanowires
• Observed in tunneling between parallel GaAs nanowires
• Results not all explained by theory
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Conclusion
• Luttinger model gives interesting phenomena
• Some experimental evidence in CNTs, nanowires (also polymers)
• Experimental situation not settled:– Spin-charge separation in CNTs?– Coulomb blockade?
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5/2/2007 Cohen Group Meeting 14
Instability of Fermi Surface